It's Not Magic, It's Mathematics

Entertainers such as magicians and mentalists incorporate numbers into their staged illusions. I am referring not to sleight of hand card tricks or other such manipulations, but to a display of mathematics camouflaged by razzle-dazzle and cries of “abracadabra”.

Although we know it is not real magic, it still seems like they are doing the impossible, just like creating impossible mathematics shapes such as the ones shown here.

This article will hopefully go some way to demystify so-called number magic and encourage you to explore the fascinating world of number patterns and algebra.

Magic 1: Is That a Zebra Crossing?

Let’s begin with one where I predict the outcome regardless of your initial choice of number.

Carry out these steps in turn, keeping track of your answer each time.

1. Think of any number.

2. Square it. That means multiply it with itself, such as 3 x 3, 8 x 8. 3. Add the result to your original number. 4. Divide the answer by your original number. 5. Add 99. 6. Subtract from the answer the number you started with. 7. Divide by 10.

8. Now add 16.

9. If A = 1, B = 2, C = 3, D = 4, etc, work out the letter that corresponds to your final answer.

10. Think of a 4-legged animal whose name starts with the letter you found.

I’m sure the animal you came up with has stripes and looks like a donkey!

Try this again using a different number. What can you conclude?

Now let’s see mathematically what is happening.

We will use the letter N to represent the start number and perform each of the 10 steps using this letter. The solution is shown alongside each step.

100xN can only have the values 200, 300, …, 900. This can be ignored in the final answer. Then (2018 – year of birth) or (2017 – year of birth) is the person’s birth year, which is obtained from the last 2 digits of the answer.

Magic 3: Hieroglyphics Prediction

This one is both interesting and easy to explain. We will use 46 as our initial number.

1. Think of a number from 10 to 99. [ 46]

2. Add its two digits together. [ 4 + 6 = 10]

3. Subtract the total from the original number. [46 – 10 = 36]

4. Find the shape next to your answer. [ a circle inside another circle]

It turns out that the answer will always correspond to a number with a circle next to it.

Let’s see why by reworking and explaining each step.

1. Suppose our 2-digit number is AB. This can be written as 10xA + B.

For example, 46 = 10x4 + 6.

2. Add the two digits together to get A + B.

3. To subtract the total from the original number, we write 10xA + B – (A + B).

This is the same as 10xA + B – A – B, which simplifies to 9xA.

Now, A is the first digit, which can be any of the digits 1, 2, 3, 4, 5,6 ,7 ,8, 9.

Therefore, 9xA are the first 9 multiples of 9.

Hence the only possible answers for choosing an initial number from 10 to 99 are 9, 18, 27, 36, 45, 54, 63, 72, 81 or 90.

If you look again at the diagram above, you will notice that the symbol next to each of these multiples of 9 is the same; a circle inside another circle.

Magic 4: Symbols Galore

This one is an interesting variation of Magic 3.

1. Choose two different digits and make a number from 10 to 99.

Suppose we choose 5 and 7 to form the number 57.

2. Reverse the two digits to get another number.

75

3. Subtract the smaller number from the larger number.

75 – 57 = 18

4. Find the symbol under your answer.

The shape is a box.

The following provides a proof that the result is always the same.

1. Suppose our two digits are A and B and we form the 2-digit number is AB.

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